On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities
Copyright policy )On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities
We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with $n$ discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the $n$ discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes $G$-function.
36 pages, 5 figures
- Sun Yat-sen University China (People's Republic of)
- Institut Franco-Chinois de l'Energie Nucléaire China (People's Republic of)
Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain, Painlevé-type functions, coupled Painlevé V system, asymptotic expansion, Random matrices (algebraic aspects), FOS: Physical sciences, Relations of finite-dimensional Hamiltonian and Lagrangian systems with algebraic geometry, complex analysis, special functions, Determinants, permanents, traces, other special matrix functions, Mathematical Physics (math-ph), Spectral theory for ordinary differential operators in the complex domain, Random matrices (probabilistic aspects), confluent hypergeometric kernel, Riemann-Hilbert approach, Fredholm determinant, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\), Asymptotics and summation methods for ordinary differential equations in the complex domain, Mathematical Physics, Nonautonomous Hamiltonian dynamical systems (Painlevé equations, etc.)
Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain, Painlevé-type functions, coupled Painlevé V system, asymptotic expansion, Random matrices (algebraic aspects), FOS: Physical sciences, Relations of finite-dimensional Hamiltonian and Lagrangian systems with algebraic geometry, complex analysis, special functions, Determinants, permanents, traces, other special matrix functions, Mathematical Physics (math-ph), Spectral theory for ordinary differential operators in the complex domain, Random matrices (probabilistic aspects), confluent hypergeometric kernel, Riemann-Hilbert approach, Fredholm determinant, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\), Asymptotics and summation methods for ordinary differential equations in the complex domain, Mathematical Physics, Nonautonomous Hamiltonian dynamical systems (Painlevé equations, etc.)
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).2 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 10% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!